Every closed curves in C/{0} are homotopic to closed curve whose trace is is contained in unit circle

Question

It's a question in Conway that is every closed curve in C/{0} is homotopic to a closed curve whose trace is contained in unit circle,any ideas or proof will be appreciated.i tried thinking but nowhere to reach.

Answer 1

Let $ \gamma(t),t\in [0,1] $ be any closed curve in $ \mathbb{C}-\{0\} $, then $$ f:[0,1]\times[0,1]\to \mathbb{C}-\{0\},(t,s)\mapsto\frac{\gamma(t)}{1+s(|\gamma(t)|-1)} $$ is the required homotopy.